= Solution
An <alpha-plane> is a totally null complex two-plane with tangent vectors $u^A\pi^{A'}$ for a fixed primed spinor. A <beta plane> has tangent vectors $\lambda^A u^{A'}$ for a fixed unprimed spinor. The <metric tensor> pairs two vectors of either family to zero, since the repeated fixed spinor contracts with an <symplectic form>. These are the two chiral families of totally null planes; labels for their self-duality depend on the orientation convention.
Globally in the <twistor correspondence>, write
$$
A_\ell=\{W:\ell\subset W\},\qquad B_H=\{W:W\subset H\},
$$
where $\ell$ is a line and $H$ a three-dimensional hyperplane in <twistor space>. Both compact families are <Complex projective spaces> $\mathbb{CP}^2$.
For distinct $\ell_1,\ell_2$, any $W\in A_{\ell_1}\cap A_{\ell_2}$ must contain their two-dimensional span. Since $W$ itself has dimension two, it must equal that span. Conversely the span belongs to both planes. Thus \b[two distinct compact alpha-planes meet in exactly one point]:
$$
\boxed{A_{\ell_1}\cap A_{\ell_2}=\{\ell_1+\ell_2\}.}
$$
In affine coordinates, independent $\pi_1,\pi_2$ determine a unique matrix $x$ from the two incidence equations. If the primed spinors are proportional while the projective twistors are distinct, no finite affine solution exists. The compact intersection still exists, at conformal infinity; saying the planes are disjoint without this affine qualification would be wrong for the printed space $M$.
Back to article page
